There is a function $f(z)$ given: $$f(z) = \frac{z}{z^2 + 1}.$$ $z \in \Omega = \mathbb{C} \setminus \overline{D(0,1)}$,
where $D(0, 1)$ is a disc centered at zero and its radius is equal to 1.
I am to check whether the given function has an antiderivative.
What am I to check?
To my mind it is necessary to check if
- integral over any line from point $a$ to point $b$ is equal to $0$.
Is this condition equivalent to the statement that $f$ is holomorphic on every single-coherent area which is included in $\Omega$?
Calculations
If $1$ is true how can I check if integrals over any line is equal to $0$?
Hint: Consider the integral of $f$ along the circle of radius $2$ centered at the origin. If $f$ has an antiderivative, then this integral is zero.
If you can, compute the integral using residues using that $$ \operatorname{Res}(f,c) = \frac{g(c)}{h'(c)} $$ when $f(z) = \dfrac{g(z)}{h(z)}$ and $h(c)=0$ and $h'(c)\ne0$.
Otherwise, compute the integral directly. Using a square instead of a circle will be simpler.